Finite groups with an automorphism of prime order whose centralizer has small rank

Let φ be an automorphism of prime order p of a finite group G, and let CG(φ) be its fixed-point subgroup. When φ is regular, that is, CG(φ)=1, the group G is nilpotent by Thompsonʹs theorem. The “almost regular” results of Fong and Hartley–Meixner–Pettet were giving the existence of a nilpotent subgroup of index bounded in terms of p and CG(φ). We prove the rank analogues of these results, when “almost regular” in the hypothesis is interpreted as a restriction on the rank r of CG(φ), and the conclusion is sought as nilpotency modulo certain bits of bounded rank. The classification is used to prove almost solubility in the coprime case: the rank of G/S(G) is bounded in terms of r and p. For soluble groups the Hall–Higman-type theorems are combined with the theory of powerful q-groups to obtain almost nilpotency, even without the coprimeness condition: there are characteristic subgroups R N G such that N/R is nilpotent and the ranks of R and G/N are bounded in terms of r and p. Examples show that our results are in a sense best-possible.